Exercise Set 2

These exercises cover the topics of Functions and Graphs.

1. Sketch the graphs of the following functions on the interval \(-2\le x \le 2\).

   1. \(y=-\frac{1}{2}(x+1)\)
   2. \(y=x^2+2\)
   3. \(y=-2x^2+2\)
   4. \(y=x^2-x-1\)
   5. \(y=3^x\)
   6. \(y=\frac{1}{x}\)

   Tips: find where the functions cross the axes; use completing the square to find the maximum or minimum of quadratics; identify any asymptotes.

2. Sketch the following graphs (with \(x\) in radians):

   1. \(y=\sin(x)\)
   2. \(y=\sin(x+\frac{\pi}{2})\) (does this look familiar?)
   3. \(y=\sin(2x)\)
   4. \(y=\sin^2(x)\)

3. If we had the graph of a function \(f(x)\), describe what would change qualitatively for the graph of \(f(a\times x)\) where \(a\) is a constant. Consider the cases:

   1. \(a>1\)
   2. \(0 < a < 1\)
   3. \(-1 < a < 0\)
   4. \(a<-1\)

4. If we had the graph of a function \(f(x)\), describe what would change qualitatively for the graph of \(f(x+b)\) where \(b\) is a constant. Consider the cases:

   1. \(b>0\)
   2. \(b<0\)

5. If we had the graph of a function \(f(x)\), describe what would change qualitatively for the graph of \(f(x)+c\) where \(c\) is a constant. Consider the cases:

   1. \(c>0\)
   2. \(c<0\)